Sharp correlations in the ARPES spectra of strongly disordered ...

Sharp correlations in the ARPES spectra of strongly disordered topological boundary modes Zohar Ringel1

arXiv:1509.08464v2 [cond-mat.mes-hall] 7 Dec 2015

1

Theoretical Physics, Oxford University, 1, Keble Road, Oxford OX1 3NP, United Kingdom.

Data from angle resolved photo-emission spectroscopy (ARPES) often serves as a smoking-gun evidence for the existence of topological materials. It provides the energy dispersion curves of the topological boundary modes which characterize these phases. Unfortunately this method requires a sufficiently regular boundary such that these boundary modes remain sharp in momentum space. Here the seemingly random data obtained from performing ARPES on strongly disordered topological insulators and Weyl semimetals is analyzed theoretically and numerically. Expectedly the disorder averaged ARPES spectra appear featureless. Surprisingly however, correlations in these spectra between different energies and momenta reveal delta-sharp features in momentum space. Measuring such correlations using nano-ARPES may verify the topological nature of the suggested weak topological insulator (Bi14 Rh3 I9 ) which thus far was not studied using ARPES due to the rough nature of its metallic surfaces.

Topological insulators (TIs) and semimetals have attracted much attention in recent years due to their novel bulk and surface properties1–3 . Their bulk band structure has certain twists characterized by robust topological numbers. These abstract topological numbers become very much real on the boundary of the material where exotic metallic phases emerge. For instance, the surface of a 3D Strong topological insulator can host a single Dirac cone with momentum and spin locked together. Similarly unconventional are the open Fermi arcs occurring on the boundaries of Weyl semimetals 3 . Such unusual phases are interesting from a fundamental point of view but also in light of potential applications. For instance spin-momentum locking may have uses in the field of spintronics 4–6 . Angle resolved photo-emission spectroscopy (ARPES) provides much of the “smoking gun” evidence 1 for topological insulators and semimetals by directly measuring the electronic dispersion on their boundaries. A serious limitation is that these boundaries must be prepared in a careful and controlled manner to avoid distorting the electronic states. Consequently one tries to cleave only along a few natural planes while others would be disordered on the atomic scale 7 and presumably too rough to do ARPES on. At least for TIs, it is clear that states remain delocalized and metallic on such disordered boundaries 1 . Still ARPES on such boundaries was so far limited to disorder which was effectively weak 8,9 . Although the cleaving problem is prevalent, one class of topological materials in which it is clear and pressing are weak TIs 10,11 . These phases can be thought of as stacks of 2D TIs and consequently not all their surfaces are metallic. For instance a surface parallel to the stacking surface will be gapped. Unfortunately, for the compound Bi14 Rh3 I9 which appears more and more likely to be a weak TI 12 , such gapped surfaces are the only natural ones to cleave 7 and so while edges of this material clearly support some in-gap states 7 verifying that these are two counter chiral edge modes has not been possible so far.

In this work the usefulness ARPES data on strongly disordered boundaries of several topological insulators and semimetals is examined analytically and numerically. Considering the ensemble of ARPES spectra generated by different disorder realizations, it is shown that while the average ARPES spectra does not contain useful information correlations within this ensemble may contain delta-function sharp features in momentum space. This occurs whenever electron transport retains its ballistic character in the face of disorder. Such is the case for integer quantum Hall effects, Chern insulators, and as shown below also Weyl semimetals. Provided time reversal symmetry (TRS) is maintained, sharp correlations also appear in 2D TIs and 3D weak TIs. For weak TIs one needs to cleave the weak TI such that it supports one or more 1D metallic edges which are decoupled from the rest thereby preventing backscattering and rendering them ballistic. In some other cases, for instance a strong 3D TI, these correlations will show a power law singularity. I also argue that using current nano-ARPES techniques 13 such disorder correlation effects may very well be within experimental reach. We begin by analyzing the following ARPES setup on the edge of a 2D TI aligned (on average) parallel to the x ˆ direction. A photon beam of energy ν is shone on the edge and emits, via the photo-electric effect, electrons out of the material. Within the simple sudden approximation the detection rate Γ(E, p~, s) for an photo-electrons of 3D momentum p~, energy E +ν, and spin s probes the density of occupied states 14 which in Green’s function formalism is given by fD (E)=Grss (E; ~k), where = denotes the imaginary part, Grss is the retarded Green’s function at crystalline momentum ~k corresponding to p~, and fD (E) is the Fermi-Dirac function. Introducing disorder on the edge one must verify that the sudden approximation still holds. The natural concern is lack of (crystalline) momentum conservation of the photo-electron due to disorder induced scattering. There are several complementary ways by which this issue may be controlled: First one may use energies

2 (ν) such that the photo-electron’s wavelength is much smaller than the dimension of detects thereby reducing the amount of scattering. Assuming detects to be a few angstroms long with a strength (V0 ) of a few eV implies that for ν = 400eV (corresponding to 0.6˚ A) scattering effects should be strongly suppressed. Alternatively one can consider lower energies and limit the disorder to be cleaving-induced such that it occurs only on a length (a) of a few angstroms from the edge. A quantitative analysis of photo-electron scattering using the Born series is given in the Supp. Mat. A. Notably even in cases when these conditions are not strictly met it is likely that only the un-scattered component of the photo-electron would contribute in a coherent and sharp manner 15 . Given that photo-electron scattering effects are indeed suppressed and repeating the standard derivations 14,16 one obtains Γ(E, p~, s) ≈

fD (E)=Grss (E; ~k, ~k).

then =hGrs (E; k)iV =

where ARPES now measures the diagonal elements in momentum space of the Green’s function in the disordered system. The continuum theory for a 2D TI edge with TRS respecting disorder is given by 17

Z

dxdy iδkn (x−y) e × L2 (5)

δ(E − svf kn )

n

 e

Ry is x V vf

 =

X

V

Z δ(E − svf kn )

dxdyeiδkn (x−y) e−

|x−y| 2l

n

where δkn = k − kn , h...iV denotes disorder averaging, and the last equality is valid for L  |x − y| and follows from standard manipulations of Gaussian integrals. Considering the limit L  l allows us to trade RL RL RL RL dx 0 dy with 2 0 dw 0 dW , (where w = (x − y)/2 0 and W = (x + y)/2) while neglecting the dependence of the region of the dW -integration on w. This yields =hGrs (E; k)iV

(1)

X

=

X

δ(E − svf kn )

n

4L−1 l−1 , 4δkn2 + l−2

(6)

and exhibits no sharp signatures in momentum space. Looking for sharp signatures we turn our attention to the following type of correlation X C(E, E 0 ; k, k 0 ) ≡ h=Gs (E; k)=Gs0 (E 0 ; k 0 )iV (7) ss0

− h=Gs (E; k)iV h=Gs0 (E 0 ; k 0 )iV . H = vf i∂x σz + V (x)

(2)

where σz is a Pauli-matrix in spin space, V (x) is the disorder potential modeled here as Gaussian with hV (x)V (y)i = l−1 vf2 δ(x − y) with l, the phase coherence length 18 , being a few atoms for strong disorder. The Green’s function obtained from this theory is related to Grss (E; ~k, ~k) is Grss (E; ~k) ≈ Grs (E; kx ),

(3)

which describes edge modes with well defined spin which are delta functions in the y-coordinates (Grs (E; k) is short for Grs (E; k, k)). While this is obviously an approximation it would be shown to capture the essential physics. Conveniently the above disorder can beR removed usi xV ing a local gauge transformation (exp(σz vf )) and the eigenstates and energies are given explicitly by 1 is ψn,s = |si × √ e L Enk ,s = svf kn

Rx

V (x) +ikn x vf

Consider the 4 wavefunction average A4 = ∗ (k)ψn,s (k)ψn∗ 0 ,s0 (k 0 )ψn0 ,s0 (k 0 )i, appearing in the hψn,s first term on the above r.h.s. Z L dwdw0 dW dW 0 i(w+w0 )(δkn +δkn0 )+(w−w0 )(δkn −δkn0 ) A4 = 4 e L4 0 (8)   R R 0 0 W +w W +w iv −1 V+ W 0 −w0 V . × e f W −w

(4)

where, s = ±1 denotes the spin orientation in the zdirection and, without loss of generality, I assumed that the edge is compact and of length L and thus supports discrete momentumReigenvalues kn = 2πn L . For simplicity 19 I also assume that V (x) = 0 . The resulting disorder averaged Green’s function is

V

To analyze the above integrals, let us split their integration region into two sub-regions (A, B). The first would be that in which the integrals [W − w, W + w] (or [W + w, W − w] if w < 0) and [W 0 − w0 , W 0 + w0 ] do not overlap and the second would be the complementary region. Within the first region, the random variR W +w R W 0 +w0 ables W −w V and W 0 −w0 V are independent. Consequently it would not contribute to any correlation. Focusing on B, we further divide it B+ and B− , according to the sign of ww0 . For ww0 > 0 the two integral terms over the random variable V add up instead of canceling. Consequently |w| and |w0 | would be effectively limited to |w|, |w0 |